Queues with stochastic service rates
- 1 January 1967
- journal article
- Published by Wiley in Naval Research Logistics Quarterly
- Vol. 14 (2) , 219-230
- https://doi.org/10.1002/nav.3800140207
Abstract
The purpose of this paper is to explore an extension of the output discipline for the Poisson input, general output, single channel, first‐come, first‐served queueing system. The service time parameter, μ, is instead considered a random variable, M. In other words, the service time random variable, T, is to be conditioned by a parameter random variable, M. Therefore, if the distribution function of M is denoted by FM(μ) and the known conditional service time distribution as B(t |μ), then the unconditional service distribution is given by B(t) = Pr {T ≤ t}. = ∫‐∞∞ B(t |μ) dFM(μ).Results are obtained that characterize queue size and waiting time using the imbedded Markov chain approach. Expressions are derived for the expected queue length and Laplace‐Stieltjes transforms of the steady‐state waiting time when conditional service times are exponential. More specific results are found for three special distributions of M: (1) uniform on [1.2]; (2) two‐point; and (3) gamma.Keywords
This publication has 3 references indexed in Scilit:
- The analysis of non-Markovian stochastic processes by the inclusion of supplementary variablesMathematical Proceedings of the Cambridge Philosophical Society, 1955
- On the Stochastic Matrices Associated with Certain Queuing ProcessesThe Annals of Mathematical Statistics, 1953
- Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov ChainThe Annals of Mathematical Statistics, 1953